Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks PDF
Preview Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks
Springer Monographs in Mathematics Leonid Bunimovich Benjamin Webb Isospectral Transformations A New Approach to Analyzing Multidimensional Systems and Networks Springer Monographs in Mathematics Moreinformationaboutthisseriesathttp://www.springer.com/series/3733 Leonid Bunimovich • Benjamin Webb Isospectral Transformations A New Approach to Analyzing Multidimensional Systems and Networks 123 LeonidBunimovich BenjaminWebb SchoolofMathematics DepartmentofMathematics GeorgiaInstituteofTechnology BrighamYoungUniversity Atlanta,USA Provo,UT,USA ISSN1439-7382 ISSN2196-9922(electronic) ISBN978-1-4939-1374-9 ISBN978-1-4939-1375-6(eBook) DOI10.1007/978-1-4939-1375-6 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2014944753 MathematicsSubjectClassification:05C82,37N99,65F30,15A18,34D20 ©SpringerScience+BusinessMediaNewYork2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ToLarissaandRebekah Foreword Thisbookprovidesanewapproachtotheanalysisofnetworksand,moregenerally, to those multidimensional dynamical systems that have an irregular structure of interactions. Here, the term irregular structure means that the system’s variables dependoneachotherindissimilarways.Forinstance,all-to-allornearest-neighbor interactionshavearegularstructure,becauseeachvariabledependsontheothersin asimilarmanner. In practice, this structure of interactions is represented by a graph, called the network’s graph of interactions or the network’s topology. Depending on the particular network, this graph may be directed or undirected, weighted or unweighted,withorwithoutloops,withorwithoutparalleledges,etc.Ineachcase, thetechniquesprovidedinthisbookcanbedirectlyappliedtothesenetworks. It is worth mentioning that although these methods are fairly new, they have already proven to be an efficient tool in some classical and more recent problems inappliedmathematics.Here,thesetechniquesarepresentedasawayinwhichto viewandanalyzereal-worldnetworks. One of the major goals of this book is to make these methods and techniques accessibletoresearcherswhodealwithsuchnetworks.Withthisgoalinmind,we notethatthecomputationsrequiredtoimplementthesetechniquesareremarkably straightforward. In fact, they can be carried out using any existing software sophisticatedenoughtoperformelementarylinearalgebra. In terms of the book’s content, we note that each of the results is given with a mathematical proof. However, with the hope that this book will be read and used as well by nonmathematicians, the text is written so that those interested in applications can safely ignore these arguments and use the stated formulas and techniquesdirectly.Still,westressthefactthatonlyabasicunderstandingoflinear algebraisneededtounderstandtheproofs. Thedefinitionsandresultswepresentaremotivatedandaccompaniedbymany examples, which both nonmathematicians and mathematicians should appreciate. vii viii Foreword Thebookalsocontainsalargenumberofexamplesandfiguresdepictingthegraphs associatedwithparticularnetworksaswellastheirvarioustransformations. Almost all of these examples deal with directed graphs. The reason is that directed graphs are more general objects than undirected graphs. However, the theory developed here works just as well for undirected graphs. This is important, forinstance,inthestudyofrealnetworks,sincealargenumberofthosenetworks haveanundirectedgraphstructure(topology). Because real-life networks are often large and have a complicated structure, it is tempting to find ways of simplifying them in terms of both their size and complexity.Whatisimportant,though,isthatsomebasicorfundamentalproperty ofthenetworkbepreservedinthisprocess.Yetsuchanattemptseemsdoomedto failure.Thesearerealnetworks,sowedonotknowmuchifanythingaboutthem, including whichcharacteristic(s)weshould retain.Moreover, therearepotentially manywaysinwhichanetworkcouldbereduced.Hence,thereisfirsttheproblemof choosing which way the network should be reduced and second determining what the reduced network tells us. Thus many objections are immediately raised if one wantstoreducethesizeofanetwork. From this point of view, our goal of reducing a network may seem overly ambitious. In fact, one could ask how it is possible even to represent an arbitrary network. The universally excepted answer is that this can be done by drawing a graphwhosevertices(nodes)correspondtothenetworkelementsandwhoseedges (links)correspondtothedirectedinteractionsbetweentheseelements. Equivalently, one can represent a network by a matrix A with entries A . In ij thisrepresentation,A isthestrengthorweightofthedirectedinteractionbetween ij theithandjthnetworkelements,whereA D 0iftheseelementsdonotinteract. ij Suchamatrixiscalledtheweightedadjacencymatrixofanetwork.Ifthenetwork’s interactionstrengthsarenotknown,thenonzeroentriesofthematrixaresetequal to1,andAiscalledthe(unweighted)adjacencymatrixofthenetwork.Inpractice, knowledgeofanetwork’sadjacencymatrixisoftenthemostonecanhopetohave. Itiswellknownthataverybasiccharacteristicofamatrixisitsspectrum,i.e., its collection of eigenvalues including multiplicities. One of the main questions we address in this book is whether it is possible to reduce a network to some smallernetworkwhilepreservingthenetwork’sspectralproperties.Phrasedanother way, this question could be stated as whether it is possible to reduce the size of a network’sadjacencymatrixwhilemaintainingthenetwork’seigenvalues. Theimmediateanswertothisquestionis,ofcourse,no.Infact,whilepresenting these results, we have had audience members protest that what we hope to do is impossible.Indeed,aseveryoneknows,thefundamentaltheoremofalgebrastates thatann(cid:2)nmatrixhasneigenvalues,whileasmallermatrixhasfewer. However, our claim is that it is possible to reduce a matrix and preserve the matrix’sspectralproperties.Inthisbook,weshowthattheanswertoourquestion becomesyesifoneconsidersalargerclassofmatrices,namelymatriceswithentries thatarerationalfunctionsofaspectralparameter(cid:2).Thatis,itispossibletoreduce an n(cid:2)n matrix with scalar entries to a smaller m(cid:2)m matrix with functions as entries and maintain the matrix’s spectrum. We refer to this process as isospectral matrixreduction. Foreword ix At this point, the reader may think that by isospectrally reducing a network’s adjacency matrix we are, in fact, shifting the complexity of a network’s structure (topology)tothecomplexityofitsedgeweights.Wepausetoreassureourreaders that we have considered this idea and that many facts and results in this book demonstrate that such is not the case. However, before moving on, we stress just onefundamentalfactregardingisospectralreductions. The structure (topology) of an isospectrally reduced network does not depend on the strengths (weights) of the initial unreduced network. It depends only on the network’s structure. The structure of the reduced network will be the same regardless of the strengths of interactions in the initial network. Therefore, the isospectral reductions we consider really capture some hidden but intrinsic informationregardingthestructureofanetwork. This approach to analyzing networks is based on ideas and methods from the theory of dynamical networks, which is a part of the modern theory of dynamical systems. The first dynamical networks addressed in this theory were the so-called coupled map lattices (CML). CMLs were introduced in the mid-1970s, almost simultaneously, by four physicists in four countries. The mathematical theory of CMLwasbegunin[7],inwhichthefirstprecisedefinitionsofspace-timechaosand acoherentstructureweregiven.Nowadays,thetheoryoflatticedynamicalsystems isarespectedpartofcontemporarydynamicalsystemstheory(see,e.g.,[13]). A number of remarkable findings of the late 1990s demonstrated that real networkshaveverycomplicatedtopologies[3,19,20,23,24,30].Thefirstthought was that the ideas of dynamical systems theory and of statistical mechanics could beappliedtosuchsystems,ashadbeendonein[7,12]forCMLs.However,infinite latticeshaveagrouptranslationproperty,whichismissingifagraphofinteractions hasanirregularstructure.Anapproachtodealingwiththisirregularstructurewas eventuallydevelopedin[1,4],inwhichthefollowingwasobserved. Every dynamical network has three features: (i) the individual dynamics of the network elements, e.g., a single isolated neuron in a neural network; (ii) the interactionsbetweentheelementsofanetwork;and(iii)thestructure(ortopology) of a network. In this framework, we assume that a network’s structure does not changeovertime,sothatithasafixedstructureofinteractions.However,aswelater pointout,thetransformationsconsideredinthisbookcouldbeusefulforstudying networksthatdohaveastructurethatevolvesovertime. Observe that features (i) and (ii) of a network are dynamical systems. Thus, it is customary to deal with such systems by analyzing the combined influence of (i)and(ii),asisdoneinotherspatiallyextendedsystems.Perhapsthemostpopular exampleoftheseisreaction–diffusionsystemsinwhichnonlinearreactionspushthe systemtowardschaoticbehaviorwhilediffusionhasastabilizingeffect.However, the question is what to do with (iii), which is clearly a static characteristic of the network. Asisshownin[1],thetopologyofanetworkcanalsobetreatedasadynamical system generated by considering all infinite paths on the network’s graph of interactions. This, together with the ideas from the theory of spatially extended systems,formsthebasisofourapproach.
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